Electromagnetic fields can often be simulated over an original domain, e.g., a surface or a volume specified geometrically. Specifically, the original domain is divided into one or more discrete sub-domains. Field-governing equations and applicable boundary conditions are modeled with a system matrix equation (e.g., a large linear system of equations that describe the behavior of a field within the discretized domain). Many approaches may be adopted for analysis of electromagnetic fields, such as the finite element method (FEM), the boundary element method (BEM), the integral equation method, and the finite different method (FDM).
For example, electromagnetic fields can be simulated using a discretized formulation of Maxwell's equations. FEM may model complex heterogeneous and anisotropic materials and represent geometrically complicated domains using, for example, tetrahedral elements, and thus is widely used to set up a system matrix equation. In essence, FEM is a numerical technique for finding approximate solutions of partial differential equations (e.g., Maxwell's equations) which often lack exact mathematical (“analytical”) solutions. FEM often represents a surface or a spatial volume as many small component elements. The discretization may be accomplished by defining a mesh grid (e.g., a triangular, tetrahedral, or other polygonal mesh) over the domain. The components of the electromagnetic field may then be expressed in a form suitable to the discretized domain. For example, the fields may be represented in a finite-dimensional function space of piecewise polynomial functions (e.g., piecewise linear functions), and the piecewise polynomial functions can be described as linear combinations of basis functions, or “finite elements.” The boundary value problem that describes the behavior of the fields in the domain (e.g., the field-governing equations and boundary conditions) is usually rephrased in a weak form, or a variational form before discretization.
Usually, FEM results in a matrix equation which may then be solved with a direct solver or an iterative solver, depending on the size and characteristics of the linear system. A direct solver corresponds to a method for directly solving a system of equations, or a computer program implementing such a method, as determined by context. For large three-dimensional problems, a direct solver may require prohibitive amounts of memory and suffer poor parallel scalability. Therefore, an iterative solver which corresponds to an iterative method for solving a system of equations or a computer program implementing such a method usually presents a practical means for solving large systems. The iterative solver often approaches the problem in successive steps, where each step refines a previous approximation to more closely approach an exact solution. A preconditioner (e.g., a matrix that reduces a condition number of the problem which in turn is a metric of the propagation of approximation errors during a numerical solution) is often applied to the original system matrix to reduce the number of necessary iterations.
The domain decomposition method (DDM) may be used to facilitate parallel solution of large electromagnetic problems and also provide an efficient and effective preconditioner. Specifically, an original domain of a problem is decomposed into several (e.g., non-overlapping and possibly repetitive) sub-domains. For example, a cuboid spatial domain may be divided into a series of smaller adjacent cubes. The continuity of electromagnetic fields at the interfaces between adjacent sub-domains is enforced through suitable boundary conditions (i.e., transmission conditions). The sub-domain boundaries may not represent actual physical boundaries, and may be introduced merely for computational convenience. Transmission conditions often specify how the fields behave at those sub-domain boundaries to ensure that the solution obtained by domain decomposition is consistent with a solution for the undivided domain (i.e., a problem formulation wherein the boundaries do not exist). For example, transmission conditions may specify that the fields, or their derivatives, are continuous across a sub-domain boundary. For problems involving electromagnetic fields, sub-domain problems are often well-posed and convergence occurs at an acceptable rate if Robin transmission conditions are imposed on the boundaries between sub-domains. Robin transmission conditions usually express these requirements in terms of particular combinations of fields, currents, and their derivatives.
However, the above-noted methods may not be sufficient alone for accurate simulation of electrically large and geometrically complicated electromagnetic problems. Instead, a hybrid method which combines a plurality of solvers may be needed. For example, for unbounded exterior problems such as an antenna radiating in free space, a hybrid finite element boundary integral (FEBI) formulation has been widely accepted as a hybrid extension to the traditional FEM method. To further extend FEM's capability to the solution of electromagnetic radiation and scattering problems involving disjoint obstacles, such as reflector antenna system, antennas mounted on large platforms, and antennas in the presence of radome structures, several methods, such as method of moments (MoM), physical optics (PO), etc., are hybridized with FEM.
A hybrid method may include the following process. An original domain is divided into one or more sub-domains. Once the solution of a sub-domain is obtained, the Huygen's equivalent sources (i.e., electric and magnetic current sources) are computed. These sources generate incident fields which subsequently impinge on all other sub-domains and make each sub-domain a well-defined scattering problem. This process continues until no significant change in the scattered fields of the sub-domains. Many hybrid methods implemented the above-noted process through a black-box approach. That is, incident fields generated by equivalent sources are represented by universal basis functions, such as nodal elements. These fields are then taken as incident fields via interpolation for each solver. Thus, the coupling of various solvers may be simple to implement requiring minor code changes. However, the black-box approach may be inherently sequential and a stationary iterative process is often used in the iteration process. Many hybrid methods may fail to handle geometrically complicated and electrically large sub-domains rigorously and efficiently.